Qhov kev hloov pauv plaub npaug yog hloov pauv los ntawm [X]=[X, X] [X]=[X, X], thiab cov kev sib txawv tuaj yeem sau tau raws li qhov sib txawv ntawm plaub qhov sib txawv los ntawm polarization tus kheej,[X, Y]=([X+Y]−[X−Y])/4.
Dab tsi yog qhov kev hloov pauv plaub ntawm Brownian motion?
Theorem 1 Lub quadratic variation ntawm Brownian motion yog sib npaug rau T nrog probability 1. |Xtk − Xtk−1 |. Yog tias peb tam sim no cia n → ∞ hauv (2) ces qhov txuas ntxiv ntawm Xt txhais tau tias qhov kev ua tsis tau zoo ntawm cov txheej txheem muaj qhov hloov pauv tag nrho thiab tsis yog xoom plaub qhov kev hloov pauv.
Puas yog quadratic variation variance?
Quadratic variation and variance is two different concepts. Cia X ua Ito txheej txheem thiab t≥0. Variance ntawm Xt yog ib qho kev txiav txim kom muaj nuj nqis uas yog qhov sib txawv ntawm plaub lub sij hawm t uas koj tau hais los ntawm [X, X]t yog qhov sib txawv tsis sib xws.
Txoj kev hloov pauv hloov pauv yog dab tsi?
Finite variation processes
Ib txheej txheem X tau hais tias muaj qhov hloov pauv tsis tu ncua yog tias nws muaj qhov sib txawv ntawm txhua lub sijhawm luv luv (nrog qhov tshwm sim 1). Cov txheej txheem zoo li no feem ntau suav nrog, tshwj xeeb, txhua qhov kev ua haujlwm sib txawv tsis tu ncua.
Puas Brownian tsab ntawv tsa suab muaj qhov hloov pauv tsis tu ncua?
Tshwj xeeb, nws qhia tau hais tias Brownian tsab ntawv tsa suab muaj nyob, qhov Brownian tsab ntawv tsa suab tsis muaj qhov sib txawv, thiab Brownian tsab ntawv tsa suab muaj finite quadratic variation.
